2 Minimization theorems and (PS) conditions

Transcription

1 Proceedings of the International Workshop Future Directions in Difference Equations. June 13-17, 2011, Vigo, Spain. PAGES Existence of solutions of discrete equations via critical point theory Alberto Cabada 1, Antonio Iannizzotto 2 and Stepan Tersian 3 1 Universidade de Santiago de Compostela, Spain 2 Università degli Studi di Catania, Italy 3 University of Ruse, Bulgaria alberto.cabada@usc.es, iannizzotto@dmi.unict.it, sterzian@uni-ruse.bg Abstract In this paper we make an overview on some classical and recent results related to the critical point theory. Moreover, we use some of the introduced results to show how to prove the existence of solutions of second order difference equations. 1 Introduction Critical point theory or Calculus of variations is a field of mathematics which deals with extremum values at points of functionals on finite or infinite dimensional spaces. Historically it goes back to brachistochrone problem, stated by J. I. Bernoulli in Acta Euditorum in June, 1696 and solved by G. Leibnitz, J. Bernoulli, G. de L Hôpital and I. Newton in The concept of finding extremum points of functionals was developed by L. Euler in Modern critical point theory started with the works of M. Morse and L. Ljusternik & L. Schnirelmann in The critical values of a functional f : X R were characterized as inf max values (inf A S max u A I (u)) over a class of sets S in the space X. Further, R. Palais (1963) [35] and R. Palais & S. Smale (1964) [36] introduced the so called (PS) condition which was a base for modern development of critical point theory. A. Ambrosetti & P. H. Rabinowitz in [2] formulated the celebrated mountain-pass theorem and applied it to solvability of nonlinear elliptic problems. In finite dimensional case it was obtained by R. Courant [19]. The concept was intensively developed by L. Nirenberg, H. Brezis, I. Ekeland, J. Mawhin, M. Willem, K. C. Chang and many other authors in different directions. During the last three decades, critical point theory has been extended in different directions and applied to various problems in differential and difference equations, mathematical physics, geometry and optimization. In the present paper we outline, in section 2, minimization and mountain-pass theorems and generalizations which are applicable to existence of solutions of boundary value problems for differential and difference equations. Such application is showed in section 3. 2 Minimization theorems and (PS) conditions A natural extension of the derivative of a function of one variable is the Fréchet derivative of a mapping in a Banach space. Let X and Y be two Banach spaces with norms X and Y 61

2 62 ALBERTO CABADA, ANTONIO IANNIZZOTTO AND STEPAN TERSIAN respectively. Let U X be an open subset and f : U Y be a mapping. When Y = R, f is said to be a functional. Definition 2.1. Let x be a point of the open subset U X. The mapping f : U Y is Fréchetdifferentiable at x U if there exists a bounded linear operator A L (X, Y ) such that f (x + h) f (x) Ah Y lim = 0. h X,h 0 h X The operator A is said to be the Fréchet derivative of the mapping f at x and denote it as Df (x) or f (x). Many equations of Mathematical physics have the operator form f (x) = 0 in an appropriate Banach space X. The equation f (x) = 0 is said to be the Euler Lagrange equation of the energy functional f : X R. Its solutions are assumed in the weak sense, i.e., < f (x), h >= 0, h X. X is the dual space of X and, is the duality pairing between X and X. The weak solutions are considered as critical points of the functional f : X R. Another derivative of a functional f is the directional derivative: the so called Gâteaux derivative of f. Definition 2.2. Let f : U Y be a mapping and x U. We say that f is Gâteaux-differentiable at x if there exists A L (X, Y ) such that f (x + th) f (x) lim = Ah, h X. t 0 t The mapping A is uniquely determined. It is called Gâteaux derivative of f at x and is denoted by D G f (x) or f G (x). We say that f is a C 1 functional, and write f C 1 (U, R), if the Fréchet-derivative f (x) exists at every point x of U and the mapping x f (x) is continuous from U to X, i.e., if lim j x j = x U then lim j f (x j ) f (x), v = 0, uniformly on {v X : v 1}. If f is Fréchet-differentiable it is clear that it is Gâteaux-differentiable. The converse is not true. However, if f : U R has a continuous Gâteaux derivative on U, then f is Fréchetdifferentiable and f C 1 (U, R), which follows from the mean value theorem. Let X be a Banach space and f : X R a functional bounded from below. A sequence (x j ) j is said to be a minimizing sequence if lim f (x j) = inf f (x). j x X The functional f : X R is said to be lower semi-continuous (respectively weakly lower semi-continuous) if whenever lim j x j = x strongly (lim j x j = x weakly), it follows lim inf j f (x j) f (x).

3 EXISTENCE OF SOLUTIONS OF DISCRETE EQUATIONS VIA CRITICAL POINT THEORY 63 The functional f : X R is sequentially weakly continuous if whenever lim j x j = x weakly, it follows lim f (x j) = f (x). j We present a criterion for weak lower semi continuity and minimization theorem (see M. Berger [5, Chapter 6] and J. Mawhin & M. Willem [33, Chapter 1]). Theorem 2.1. Let X be a reflexive Banach space and f : X R be a functional represented as the sum f = f 1 +f 2, where f 1 is continuous and convex and f 2 is sequentially weakly continuous. Then f is weakly lower semi-continuous. Theorem 2.2. Let f be a weakly lower semi-continuous functional on the reflexive Banach space X with a bounded minimizing sequence. Then f has a minimum on X. Now we introduce the concept of coercive functional Definition 2.3. A functional f is said to be coercive if f (x) + as x. By the last theorem we have Corollary 2.3. Let f be a weakly lower semi-continuous functional bounded from below on the reflexive Banach space X. If f is coercive, then c = inf f is attained at a point x 0 X. Note that if X is separable and f is sequentially weakly lower semi-continuous then the same conclusion holds. Let M be a complete metric space and Φ : M R be a lower semi-continuous functional, bounded from below. If (u j ) j is a minimizing sequence, then for every ε > 0 there exists j 0 such that for j > j 0 Φ (u j ) inf M Φ + ε. We say that u is an ε minimum point of Φ if Φ (u) inf M Φ + ε. Ekeland theorem [21] considers the existence of ε minimum points. Theorem 2.4 (Ekeland Principle, strong form, 1979). Let M be a complete metric space and Φ : M R be a lower semi continuous functional which is bounded from below. Let k > 1, ε > 0 and u M be an ε minimum point of Φ. Then there exists v M such that Φ (v) Φ (u), d (u, v) 1 k, Φ (v) < Φ (w) + ε k d (w, v), w v.

4 64 ALBERTO CABADA, ANTONIO IANNIZZOTTO AND STEPAN TERSIAN Minimizing sequences for differentiable functionals are convergent under certain compactness conditions. We shall use later the so called Palais Smale condition (PS-condition for short), appeared in the book of J. T. Schwartz [43] as Definition 2.4. A C 1 -functional f : X R satisfies the Palais Smale (PS) condition if any sequence (x j ) j in X such that f (x j ) is bounded and lim j f (x j ) X = 0 has a convergent subsequence. From (P S) condition, it follows that the set of critical points for a bounded functional is compact. A variant of (P S) condition, noted as (P S) c, was introduced by H. Brézis, J. M. Coron and L. Nirenberg [9]. Definition 2.5 (Brézis, Coron, Nirenberg, 1980). Let c R be given. A C 1 functional f : X R satisfies the (P S) c condition if any sequence (x j ) j in X such that lim j f (x j ) = c and lim j f (x j ) X = 0 contains a convergent subsequence. It is clear that the (P S) condition implies the (P S) c condition for every c R. Moreover, the (P S) c condition implies the compactness of the set of critical points at a fixed level c. A consequence of Theorem 2.4 for differentiable functionals is the following Theorem 2.5. Let f : X R be a C 1 functional bounded from below. Then, for each ε > 0 and x X such that there exists y X such that By last theorem it follows f (x) inf X f + ε, f (y) f (x), x y ε, f (y) X ε. Corollary 2.6. Let f : X R be a C 1 functional bounded from below and (x j ) j be a minimizing sequence. Then there exists another minimizing sequence (y j ) j such that f (y j ) f (x j ), lim x j y j = 0, j lim f (y j ) X = 0. j Now, combining this result with (P S) condition, we get Theorem 2.7. Let f : X R be a C 1 -functional bounded from below and c = inf f. Assume that f satisfies (P S) c condition. Then c is achieved at a point x 0 X and f (x 0 ) = 0. The last theorem has a generalization based on another (P S) condition introduced by G. Cerami [17].

5 EXISTENCE OF SOLUTIONS OF DISCRETE EQUATIONS VIA CRITICAL POINT THEORY 65 Definition 2.6 (Cerami, 1978). Let c R. We say that the C 1 -functional f : X R satisfies the (P SC) c condition if every sequence (x j ) j in X such that has a convergent subsequence. lim f (x j) = c and lim (1 + x j ) f (x j ) X = 0, j j Theorem 2.8. Let f : X R be a C 1 -functional bounded from below and c = inf f. If (P SC) c condition is satisfied, then c is a minimum of f. 3 Mountain pass and three critical points theorems The original approach to prove the Mountain Pass Theorem (MPT) of A. Ambrosetti & P. H. Rabinowitz [2] is based on the Deformation Theorem. Let f C 1 (X, R) be a functional defined on the open subset X in the Banach space E. We introduce the following notations f c = {x E : f (x) c}, f c = {x E : f (x) c}, f b a = {x E : a f (x) b} = f a f b, K c = { x E : f (x) = c, f (x) = 0 }. A continuous mapping η : (t, x) [0, 1] X η (t, x) X is said to be a homotopy of homeomorphisms if (1) η (t, x) : X X is a homeomorphism for all t [0, 1], (2) η (0, x) = x, for all x X. The following theorem is proved by P. H. Rabinowitz (see [38, Appendix A, Theorem A.4]). Theorem 3.1 (Deformation Theorem). Let f C 1 (E, R) satisfy the (PS) condition. If c R, ε 0 > 0 and N is any neighborhood of K c, then there exists an ε (0, ε 0 ) and η C ([0, 1] E, E) such that (1) η (0, x) = x, for all x E ; (2) η (t, x) = x, for all t [0, 1] if f (x) c ε 0 ; (3) η (1, f c+ε \N) f c ε ; (4) if K c = then η (1, f c+ε ) f c ε ; (5) if f is even, then η (t, x) is odd in x for all t [0, 1]. Theorem 3.1 was generalized by M. Willem [46, 47], P. Bartolo, V. Benci & D. Fortunato [4], H. Brezis & L. Nirenberg [10] in various directions. We recommend to the reader the books on the topic for further studies, where detailed proofs of theorems and their applications are presented (see, among others, [26, 29, 33, 38, 47, 49]).

6 66 ALBERTO CABADA, ANTONIO IANNIZZOTTO AND STEPAN TERSIAN Theorem 3.2 (Mountain Pass Theorem, Ambrosetti & Rabinowitz, 1973). Let X be a real Banach space and f C 1 (X, R). Suppose that f satisfies the (P S) condition, f (0) = 0 and (i) there exist constants ρ > 0 and α > 0 such that f (x) α if x = ρ ; (ii) there is e X, e > ρ, such that f (e) 0. Then f has a critical value c α which can be characterized as where c = inf max f (γ (t)), (1) γ Γ t [0,1] Γ = {γ C ([0, 1], X) : γ (0) = 0, γ (1) = e}. (2) Geometrically, when X = R 2 the assumptions (i) and (ii) mean that the origin lies in a valley surrounded by a mountain Γ f = { (x, f (x)) R 3 : x R 2}. So, there must exist a mountain pass joining (0, 0) and (e, f (e)) that contains a critical value. Note that the (P S) condition is essential in Theorem 3.2. The function h (x, y) = x 2 + (x + 1) 3 y 2 satisfies assumptions (i) and (ii) of Theorem 3.2, but does not satisfy the (P S) condition and its unique critical point is (0, 0). Following weaker form of mountain-pass theorem is due to D. G. Figueredo & S. Solimini (see [23, 24]). Theorem 3.3. Let f C 1 (X, R) satisfying the (P S) condition. Suppose that inf{f(x) : x = r} max{f (0), f (e)}, where 0 < r < e. Then f has a critical point x 0 0. Corollary 3.4. Let f C 1 (X, R) satisfying the (P S) condition. Suppose that f has two local minima. Then f has at least one more critical point. The next result is due to P. H. Rabinowitz [38]. Theorem 3.5 (Linking theorem, Rabinowitz, 1986). Let X = X 1 X2 be a real Banach space, where X 1 {0} is finite dimensional. Suppose that f C 1 (X, R), satisfies the (PS) condition and (I 1 ) There exist constants σ, ρ > 0 such that sup{f(x) : x = ρ, x X 1 } σ, (I 2 ) There exist e X 1, with e = ρ and a constant ω > σ such that inf{f(e + x) : x X 1 } ω. Then f possesses a critical value c 0 and where c = inf γ Γ max {f (γ (u)) ; u X 1, u = ρ}, Γ = {γ C (B[0, ρ] X 1, X) such that γ(u) = u, for all u B[0, ρ] X 1 }.

7 EXISTENCE OF SOLUTIONS OF DISCRETE EQUATIONS VIA CRITICAL POINT THEORY 67 The last statement infers the existence of three critical points of a functional. Other three critical points theorems with (P S) condition were proved by P. Pucci & J. Serrin [37], J. Mawhin & M. Willem [33], V. Moroz, A. Vignoli & P. Zabreiko [34], B. Ricceri [39]. We point out the following Ricceri s theorem which has several generalizations (see [6, 11]) and applications to differential and difference equations (see [12, 20, 40]). Theorem 3.6 (Ricceri [39]). Let X be a separable and reflexive real Banach space, I R an interval, and g : X I R a continuous function satisfying the following conditions: (i) for each x X the function g(x, ) is concave; (ii) for each λ I, the function g(, λ) is sequentially weakly lower semi continuous, coercive and continuously Gâteaux differentiable, satisfies the (PS) condition; (iii) there exists a continuous concave function h : I R such that sup λ I inf x X (g(x, λ) + h (λ)) < inf sup x X λ I (g(x, λ) + h (λ)). Then, there exists an open interval Λ I and a positive real number σ such that for each λ Λ the equation g x (x, λ) = 0 admits at least three solutions in X whose norms are less than σ. A recent consequence of Ricceri s theorem due to A. Cabada & A. Iannizzotto [11] as follows Theorem 3.7. Let (X, ) be a uniformly convex Banach space with strictly convex dual space, J C 1 (X, R) be a functional with compact derivative, x 0, x 1 X, p, r R be such that p > 1 and r > 0. Assume that the following conditions are satisfied: J(x) (i) lim inf x x 0; p (ii) inf J(x) < inf J(x); x X x x 0 r (iii) x 1 x 0 < r and J (x 1 ) inf J(x). x x 0 =r Then, there exists a nonempty open set A (0; + ) such that for all λ A the functional has at least three critical points in X. x x x 0 p p + λ J(x) To finish this section, let us note also variants of Mountain Pass Theorem due to H. Brezis & L. Nirenberg [10] without (PS) condition. Let Q be a compact metric space and let Q be a nonempty closed subset Q and p be a fixed continuous map on Q. Define and so that A := {p C (Q, X) : p = p on Q } c := inf max p A t Q f (p (t)), c max t Q f (p (t)).

8 68 ALBERTO CABADA, ANTONIO IANNIZZOTTO AND STEPAN TERSIAN Theorem 3.8 (Brezis & Nirenberg, Standard MPT). Let f C 1 (X, R). Assume that for every p A, max t Q f (p (t)) is attained at some point in Q\Q. Then there exists a sequence (x j ) j in X such that lim f (x j) = c and lim f (x j ) = 0. j j In addition, if f satisfies the (P S) c condition, then c is a critical value. Moreover, if (p j ) j is any sequence in A such that then there exists a sequence (t j ) j in Q such that c = lim j max t Q f (p j (t)), lim f (p j (t j )) = c and lim f (p j (t j )) = 0. j j Theorem 3.2 is clearly a special case of Theorem 3.8 with Q = [0, 1], Q = {0, 1} and p (t) = te. The next Theorem of N. Ghoussoub [25] (see [10]) contains an earlier result of P. Pucci & J. Serrin [37] as a special case. Theorem 3.9. Assume the conditions of Theorem 3.8 and that there is a closed set Σ X, disjoint from p (Q ), on which f c and such that for all p A, p(q) Σ. Then there exists a sequence (x j ) j in X such that lim f (x j) = c, j lim f (x j ) = 0, and lim dist (x j, Σ) = 0. j j 4 Existence results for difference equations As we have noticed in the previous sections, the variational methods coupled with critical point theory have been extensively used in the literature to deduce existence results for Ordinary and Partial Differential Equations, see for instance the monographs of H. Brezis [8], M. R. Grossinho & S. Tersian [26] and S. Kesavan [32]. However, if we refer to difference equations, this application is relatively new. Some of the first steps on the literature related to this field are referred on the paper by Z. Guo & J. Yu [27], in which the authors survey some results concerning the existence and multiplicity of solutions for several kinds of difference equations. In particular they consider the second order non autonomous difference equation 2 x n 1 + f(n, x n ) = 0, n Z, (3) in which f C(R R, R) such that f(n + M, s) = f(n, s) for every n N, s R and some M N. By assuming, among other conditions, that f(t, ) is superquadratic at infinity, the authors prove the existence of at least one nontrivial periodic solution of period p M (for a given p N). They prove that the p M periodic solutions of this problem coincide with the critical points of the functional p M ( ) 1 J(x) = 2 ( x n) 2 + F (n, x n ), x R p m, n=1

9 EXISTENCE OF SOLUTIONS OF DISCRETE EQUATIONS VIA CRITICAL POINT THEORY 69 where F (n, s) s 0 f(n, τ) dτ, n Z, s R. The proofs are deduced from the fact that operator J satisfies the (PS) condition and by applying the (linking) Theorem 3.5. Using a more sophisticated argument, the same authors prove the existence o three periodic solutions in [28]. The vectorial case has been considered by Z. Zhou, J. Yu & Z. Guo in [50]. Equation (3), coupled with Dirichlet boundary conditions, has been studied by R. P. Agarwal, K. Perera & D. O Regan in [1]. The authors deduce the existence of at least two nonnegative solutions by assuming some suitable growth conditions on the asymptotic behavior of function f at 0 and at +. The results follow from the (mountain pass) Theorem 3.2. A generalization of this results for higher dimensional systems has been done by L. Jiang & Z. Zhou in [30]. Also, F. Faraci & A. Iannizzotto obtained some multiplicity results for discrete Dirichlet problems in [22]. The self adjoint second order equation (p n x n 1 ) + q n x n = f(n, x n ), n Z, has been studied by J. Yu, Z. Guo & X. Zou in [45], with periodic boundary conditions, and by Y. Li in [31] for the Dirichlet ones (with q 0 and f(n, x) g(n) h(n, x)). Problems with parametric dependence have been considered by G. Zhang & S. Liu in [48]. In particular, they study the autonomous Dirichlet problem 2 x n 1 + λf(x n ) = 0, n {1,..., T }, x(0) = x(t + 1) = 0. (4) By means of the (mountain pass) Theorem 3.2, the authors prove that if there is a > 0 such that f(0) = a and F (δ) δ 0 f(s) ds > 0 for some δ > 0, then problem (4) has at least one positive solution on {1,..., T } for all λ > 0, provided that function f is sublinear at + : and there are θ (0, 1/2) and R > 0 such that f(u) lim u + u = 0, F (s) θs f(s), for all s R. Moreover they prove that if f is sublinear at 0: f(u) lim u 0 u = 0, then there are 0 < Λ 0 Λ 1, for which problem (4) has no positive solution for all λ (0, Λ 0 ) and two positive solutions for all λ > Λ 1. The analogous non autonomous Dirichlet problem 2 x n 1 + λf(n, x n ) = 0, n {1,..., T }, x(0) = x(t + 1) = 0, (5)

10 70 ALBERTO CABADA, ANTONIO IANNIZZOTTO AND STEPAN TERSIAN has been studied by D. Bai & Y. Xu in [3]. On this work, the authors prove that if function f is continuous and odd in the second variable and there exists α > 0 such that f(n, α) = 0 and f(n, u) > 0 for u (0, α), then there exists λ > 0 such that problem (5) has at least T distinct pairs of nontrivial solutions for all λ > λ. Furthermore, each nontrivial solution u satisfies that u n α for all n {1,..., T }. The proof follows from the following multiplicity result, known as Clark s theorem, given by P. H. Rabinowitz in [38]. Theorem 4.1. Let X be a real Banach space, f C 1 (X, R) even, bounded from below, and satisfying the (PS) condition. Suppose f(0) = 0, there is a set K Σ such that K is homeomorphic to S j 1 (the j 1 dimension unit sphere) by an odd map, and sup {f(x) : x K} < 0. Then the functional f has at least j distinct pairs of nonzero critical points. The p laplacian equation (with ϕ p (s) = s p 2 s, p > 1) has been considered by P. Chen & H. Fang in [18]. On that paper the authors studied the general second order equation (ϕ p ( x n 1 )) + f(n, x n+1, x n, x n 1 ) = 0, n Z. By assuming the continuity of the function f and its periodicity on the first variable, using the (linking) Theorem 3.5, they obtain a sufficient condition for the existence of periodic and subharmonic solutions of the previous equation. The parameter dependence on the p laplacian equation has been considered by G. Bonanno & P. Candito in [7]. They study the following problem, which generalizes (5): (ϕ p ( x n 1 ) + λf(n, x n ) = 0, n {1,..., T }, x(0) = x(t + 1) = 0. (6) The authors obtain some real intervals in which for λ in such intervals, problem (6) has at least three distinct positive solutions. To deduce these multiplicity results they impose some growth assumptions on the behavior of the function f at 0 and/or at +. The critical point result employed in this case is the following, proved in [7, Theorem 2.1]: Theorem 4.2. Let X be a finite dimensional real Banach space and I λ : X R be a functional such that I λ (u) := Φ(u) λψ(u) for all u X. Here Φ, Ψ : X R are two functions of class C 1 on X, with Φ coercive, and λ is a positive real parameter. Assume, moreover that 1. Φ is convex and inf X Φ = Φ(0) = Ψ(0) = For each λ > 0 and for every x 1, x 2 which are local minima for the functional Φ λψ and such that Ψ(x 1 ) 0 and Ψ(x 2 ) 0 one has inf Ψ(tx 1 + (1 t)x 2 ) 0. t [0,1] Further, assume that there are two positive constants ρ 1, ρ 2 and v X, with ρ 1 < Φ( v) < ρ 2 /2, such that sup u Φ 1 (,ρ 1 ) Ψ(u) ρ 1 < Ψ( v) 2 Φ( v) ; sup u Φ 1 (,ρ 2 ) Ψ(u) ρ 1 < Ψ( v) 4 Φ( v).

11 EXISTENCE OF SOLUTIONS OF DISCRETE EQUATIONS VIA CRITICAL POINT THEORY 71 Then, for each λ ( 2Φ( v) Ψ( v), min { }) ρ 1 sup Ψ(u), ρ 2 /2 u Φ 1 (,ρ1 ) sup u Φ 1 (,ρ2 ) Ψ(u) the functional I λ admits at least three distinct critical points u 1, u 2, u 3 such that u 1 Φ 1 (, ρ 1 ), u 2 Φ 1 (ρ 1, ρ 2 /2) and u 3 Φ 1 (, ρ 2 ). Different results in this direction have been given by P. Candito & G. D Agui [15] and P. Candito & N. Giovannelli [16]. In [12], we prove the following multiplicity result for the solutions of the problem (6). Theorem 4.3. Let r, s R satisfying 0 < r < s. Assume that the following conditions hold: F (k, t) 1. lim sup t + t p 0 for every k [1, T ]; 2. T sup k=1 t s c 1 F (k, t) < T k=1 3. sup F (k, t) r t s c 2 c 1 sup F (k, t); t R sup h k t s c 1 F (h, t) for every k [1, T ]. Then, there exists λ > 0 such that (6) admits at least three solutions. Here c 1 and c 2 denote the embedding constants of the solutions space into R T, endowed with the maximum norm (see [12] for the exact computation of such constants). The proof follows by combining the (mountain pass) Theorem 3.2 together with the following theorem of Ricceri (see [42, Theorem A] or [41, Theorem 1]). Theorem 4.4. Let (X, τ) be a Hausdorff space and Φ, J : X R be functionals; moreover, let M be the (possibly empty) set of all the global minimizers of J and define β = α = inf x X Φ(x), inf Φ(x) if M x M Φ(x) if M =. sup x X Assume that the following conditions are satisfied: 1. for every µ > 0 and every ρ R the set {x X : Φ(x) + µj(x) ρ} is sequentially compact (if not empty); 2. α < β. Then, at least one of the following conditions holds: (a) there exists a continuous mapping h : (α, β) X with the following property: for every t (α, β), one has and for every x Φ 1 (t), x h(t) Φ(h(t)) = t J(x) > J(h(t));

12 72 ALBERTO CABADA, ANTONIO IANNIZZOTTO AND STEPAN TERSIAN (b) there exists µ > 0 such that the functional Φ + µ J admits at least two global minimizers in X. In [12] it is pointed out that while the first condition on Theorem 4.3 is a standard coercivity assumption, the next two ones are rather unusual; however such assumptions cannot be, in general, removed or weakened. Theorem 4.3 can be improved using Theorem 3.7: Theorem 4.5. Assume that the following conditions are satisfied: (i) lim sup t (ii) F (k, t) t p 0 for all k [1, T ]; T sup F (k, t) < k=1 t s T k=1 sup F (k, t) for some s > 0; t R (iii) there exists ξ R T such that ξ < s and T F (k, ξ k ) = sup k=1 η s k=1 T F (k, η k ). Then, there exists a nonempty open set A (0, + ) such that for all λ A problem (6) admits at least three solutions. In [14], A. Cabada & S. Tersian prove the existence of heteroclinic solutions of the equation (ϕ p ( x n 1 ) + λf(n, x n ) = 0, n {1,..., T }, x( ) = 1; x(+ ) = 1. The proof follows by constructing a sequence of solutions of related Dirichlet problems, and from the application of some existence results of P. Candito and N. Giovannelli in [16]. In [13], by means of the (mountain pass) Theorem 3.2, A. Cabada, C. Li & S. Tersian prove the existence of homoclinic solutions for the p Laplacian difference equation (ϕ p ( x n 1 ) V n ϕ q (x n ) + λf (n, x n ) = 0, x n 0, n. (7) Here functions V and f(, x) are T -periodic and satisfy the following assumptions: The potential function F (n, t) of f (n, t) satisfies the Rabinowitz s type condition: (F 1 ) There exist µ > p q > 1 and s > 0 such that µ F (k, t) t f (k, t), k Z, t 0 and F (k, t) > 0, k Z, for t s > 0. ( (F 2 ) f (k, t) = o t q 1) as t 0. Let Φ q (t) = t q q. The main obtained result is the following Theorem 4.6. Suppose that the function V : Z R is positive and T -periodic and the functions f (k, ) : Z R R satisfy assumptions (F 1 ) (F 2 ). Then, for each λ > 0, the equation (7) has a nonzero homoclinic solution x l q, which is a critical point of the functional J : l q R: J (x) = n Z Φ p ( x (n 1)) + n Z V (n) Φ q (x (n)) λ n Z F (n, x (n)). Moreover, given a nontrivial solution x of problem (7), there exist two integer numbers k ± such that for all k > k + and k < k, the sequence x(n) is strictly monotone.

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